Generalized Eigenvalues of Pauli Matrices

  • #1
thatboi
127
18
Consider a generic Hermitian 2x2 matrix ##H = aI+b\sigma_{x}+c\sigma_{y}+d\sigma_{z}## where ##a,b,c,d## are real numbers, ##I## is the identity matrix and ##\sigma_{i}## are the 2x2 Pauli Matrices. We know that the eigenvalues for ##H## is ##d\pm\sqrt{a^2+b^2+c^2}## but now suppose I have the matrix ##\tilde{H} = (\sigma_{x} \otimes A) + (\sigma_{y} \otimes B) + (\sigma_{z} \otimes C)## , where ##\otimes## denotes the Kronecker product and ##A,B,C## are now N x N diagonal matrices with diagonal entries ##a_{i},b_{i},c_{i}## respectively. I'm wondering if there is some nice generalization of the 2x2 eigenvalue formula in my first statement? I feel like there must be.
 
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  • #2
I did some half-baked thinking and scratch work and think it is tractable. Seems like the answer is ##c_i \pm \sqrt{a_i^2+b_i^2}##. The matrix can be written as something that looks like:

##\tilde{H}=diag(a_1\sigma_x+b_1\sigma_y+c_1\sigma_z, \ldots,a_N\sigma_x+b_N\sigma_y+c_N\sigma_z)##

So it is just a bunch of copies of the simple case.
 

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